A boat that can travel at 4.0 km/hr in still water crosses a river with a current of 1.8 km/h. at what angle must the boat be pointed upstream to travel straight across the river?
Confused on how to even set it up!
To determine the angle at which the boat must be pointed upstream to travel straight across the river, we can use trigonometry. Let's break down the problem into smaller steps:
Step 1: Understand the problem and identify the given information.
- The boat can travel at 4.0 km/hr in still water.
- The river has a current of 1.8 km/hr.
Step 2: Define the variables.
- Let's assume the angle at which the boat is pointed upstream is θ.
- Let's represent the boat's speed in still water as Vb and the velocity of the river's current as Vc.
Step 3: Set up the equations.
- The boat's velocity (Vb) is made up of two components: the component that acts in the direction of its intended path (Vbθ) and the component that acts perpendicular to its intended path (Vc).
- Using this information, we can set up the following equations:
Vbθ = Vb * cosθ
Vc = Vb * sinθ
Step 4: Apply the given information.
- We know that the boat's speed in still water is 4.0 km/hr and the river's current is 1.8 km/hr, so we can substitute these values into the equations:
Vb = 4.0 km/hr
Vc = 1.8 km/hr
Step 5: Solve the equations.
- Now, we can substitute the given information into the equations and solve for θ:
Vbθ = Vb * cosθ
4.0 * cosθ = 1.8
- Divide both sides of the equation by 4.0 to isolate cosθ:
cosθ = 1.8 / 4.0
- Use the inverse cosine function (cos^-1 or arccos) to find the angle θ:
θ = cos^-1(1.8 / 4.0)
Step 6: Calculate the angle.
- Input the expression into a calculator or use a trigonometric table to find the angle:
θ ≈ 63.4 degrees
Therefore, the boat must be pointed at an angle of approximately 63.4 degrees upstream to travel straight across the river.
To solve this problem, we can use vector addition. Let's break down the boat's velocity into two components: one parallel to the river and the other perpendicular to the river.
The downstream speed of the boat with respect to the ground can be calculated by adding the boat's still water speed (4.0 km/hr) to the current speed (1.8 km/hr). So, the downstream speed is the hypotenuse of a right triangle with sides 4.0 km/hr and 1.8 km/hr.
To find the boat's speed upstream, we need to subtract the current speed from the still water speed. So, the upstream speed is the remaining side of the right triangle, which can be found by using the Pythagorean theorem.
Now, to determine the angle at which the boat must be pointed, we need to consider the relationship between the components of velocity in a right triangle. The angle opposite to the perpendicular component of velocity (upstream speed) is the angle we need to find.
Let's calculate the upstream speed and then determine the angle:
Step 1: Find the downstream speed:
Downstream speed = Boat's still water speed + Current speed
= 4.0 km/hr + 1.8 km/hr
= 5.8 km/hr
Step 2: Find the upstream speed:
Upstream speed = Boat's still water speed - Current speed
= 4.0 km/hr - 1.8 km/hr
= 2.2 km/hr
Step 3: Calculate the angle:
Using trigonometry, we can use the tangent function to find the angle:
Tangent of the angle = perpendicular component of velocity / parallel component of velocity
Tangent of the angle = (upstream speed) / (downstream speed)
angle = arctan((upstream speed) / (downstream speed))
Plugging in the values:
angle = arctan(2.2 km/hr / 5.8 km/hr)
Using a calculator or trigonometric table, you'll find the angle to be approximately 21.1 degrees.
Therefore, the boat must be pointed approximately 21.1 degrees upstream to travel straight across the river.